Blackbody Radiation

Jonathan Osbourne

Blackbody radiation states that every object at a finite temperature radiates light. Hotter objects emit more light and at smaller wavelengths when compared to cooler objects. Sefan-Boltzmann displacement law and Wien's displacement law are ways that we can calculate an objects' temperature.

Let's talk about the blackbody radiation spectrum. this was one of the most important things, well, a lot of important things came out of the nineteenth century, but this was a very very very important thing that came out of the nineteenth century.

So here's the idea. When things are in thermal contact, what that means is that they're bumping into each other, having collisions and sharing energy. So if I've got a hot liquid and a cold liquid and I pour them together, then the hot liquid is going to give some of its energy to the cold liquid, so the cold liquid will get hotter and the hot liquid will get colder until they're at the same temperature. And that's what thermal contact does. Now when they get into the same temperature, that's thermal equilibrium. The easy way to think about thermal equilibrium is everybody gets his share. Alright?

So when you're in thermal equilibrium, all the atoms get the same amount of energy provided that they're using it. There's all these little rules that I don't want to talk about right now but, you can think about it that way. So, what about light? I mean if I'm talking about atoms and they're interacting with each other, atoms consist of electrons and nuclei and those things are charged. So, they ought to interact with the electromagnetic field so that means that they ought to be able to give some energy to electromagnetic waves. So doesn't light get to play? Yes. Light gets to play and that's a major major thing because what that means is that every thing that has a finite temperature, everything, in order to be in thermal equilibrium, it's got to give some of it's energy to light. And of course, what does light do when you give it energy? It leaves. If this is going to sit around, it's not, you know attracted to the matter. It doesn't care, it's just going off. It's like a light bulb. You turn it on, well light leaves, right?

So here's the idea. Every object that has a finite temperature is always radiating energy and this energy just comes from the thermal equilibrium. It comes from the temperature itself. So it's just radiating, I'm radiating energy right now, light energy. So when we go ahead and measure the intensity of the light that's coming off versus the wavelength, we get this real interesting curves. The red curve is for a hot object, the blue curve is for a cooler object. Notice that the red curve has a lot more energy, a lot more intensity. So that means that hotter objects give up a lot more light than cooler objects still. Alright? Similarly, we find that there's a little peak here. Notice that the cooler object peaks at a higher wavelength than the hotter object does. So what that means is that as an object gets hotter, it not only radiates more just overall, it also starts to radiate at lower wavelengths.

So here, this one wasn't radiating very much at that wavelength before, but then we heat it up and now that becomes the maximum radiation. Alright. So that's the way that blackbody radiation works. These curves are extremely important. They're actually lead Planck to propose photons. That's where photons come from. They come form this blackbody radiation spectrum. Now, you can also understand this just kind of qualitatively, if you've ever seen an electric range, you turn it on, turn it up to high and what happens? How do you know it's hot? Well, the filament is glowing. right? It starts glowing red. So that's an invitation that it's hot. That means that the wavelength of peak emission has moved over far enough that you can actually see visible light. See, at my temperature right now, I'm still radiating but I'm not radiating in the visible spectrum. It's got to be pretty high to radiate in the visible spectrum. alright. Let's get qualitative on this. There's two major major major formulas that are associated with the blackbody radiation formula.

Now, I'm not going to write out Planck's formula because it's kind of complicated. But these are two things that are derived form it. The first one is the Stefan Boltzmann law and that tells you how much energy per unit time. How much power is emitted by something that has temperature t. So it tells us that power is equal to sigma, sigma is the Stefan Boltzmann law constant. And you can measure it or you can calculate it from fundamental constant, it's 5.67 times 10 to the -8 watts per meter squared kelvin to the fourth, alright? Now, that unit of course you can get just by saying the power has to be watts and look what I'm multiplying by. This is the area, the surface area of the object. Obviously if it's bigger, it's going to radiate more. I mean think about the earth, right? The earth is radiating through every little piece of its surface area. So it's a lot bigger than a beach ball. So I would expect it to radiate a lot more energy. So that's why the area is there.

Then we have this e, this is called the emissivity and you can basically think about it in terms of like greenhouse gases. Right? The earth emits energy but then some of it bounced back so that would make the amount that it's actually emitting, less, okay? So this is the emissivity alright? It's between zero and one. Alright? And then we have the temperature and what's weird about this is that the temperature is to the fourth power. Most of the time we just see things squared. At best cubed. But this is temperature to the fourth power. And it's also the kelvin temperature. It is not degrees celsius. And it's not degrees Fahrenheit certainly. It's kelvin temperature.

Now, as you remember from chemistry, kelvin temperature is celsius temperature plus 273. Now what's interesting about that is that room temperature's about 300 kelvin. So when you look at this number and you say, oh it's 10 to the -8, that's not very much power at all. Think about the fact that you are then going to multiply it by 300 to the fourth power, alright? So that actually makes this power fairly large. Anyway, so that's the first equation.

The second one is Wien's displacement law. Now this tells us how we can calculate the wavelength associated with maximum intensity if we know the temperature. So he tells us that lambda max is equal to 2.898 millimeter kelvin and that's just a constant you could measure it in experiment divided by the temperature. Again that temperature has to be in kelvin. Alright.

So let's just go ahead and do some examples. So a human being has a body temperature, internal body temperature of 98.6 degrees Fahrenheit. That translates to 310 kelvin. So Wien's displacement law tells us that the maximum wavelength is 9.35 micrometers. So this is a microwave. You can't see it, it's not visible. It's in the far infrared, alright? But, that's how thermal imaging works. And that's how those heat goggles work. You look at something and you're looking at just part of the spectrum and at this part of the spectrum, all human beings are like light bulbs. They're just radiating. Alright. So we've got that for a human. Let's get hotter.

What about molten iron? Now, you guys know that when you heat up iron enough that it starts to melt, it starts to show that red. It starts to glow. Right? So the temperature at which iron melts is 1810 kelvin. Using Wien's displacement law we see that the maximum wavelength of emission is at 1600 nanometers. Now that's not visible. It's infrared. The highest wavelength that we can see as human beings is about 700 nanometers. So this is a little bit more than twice that. However, as you remember from the curves, the maximum is not the only wavelength that's being emitted. It emits at smaller wavelengths too. And so it's just kind of beginning to glow that red. That's the idea because that red is about 650, 670 nanometers. Alright. So that's molten iron.

What about the sun? The sun has a surface temperature of about 5800 kelvin. That's very hot and when we use Wien's displacement law, we find a maximum wavelength emission of 500 nanometres. That's like a blue green. So now you might ask, well jeez. I look at the sun, it doesn't look like blue green to me. It looks yellow. There's many reasons for that. Part of the reason is that when you look at the sun you are actually seeing that whole spectrum, and there's a lot more of the larger wavelengths than they are of the smaller ones. So you're averaging in all those reds and all those oranges and all those things like that. And that pulls what it looks like kind of further up the wavelength spectrum. The other reason is that when you're looking up at the sun, you're really only seeing the light that hasn't been scattered away by the atmosphere. If you look away from the sun, what do you see? You see blue. You see the sky. That's because the majority of the light that's scattered by the atmosphere is the small wavelength stuff. So that stuff's going away. What's left over? The bigger wavelengths and that's why the sun appears yellow when you look at it. Alright.

And there's one other example that I wanted to give. This is the cosmic background radiation which is something that's very very very important to physicists all around the world. Essentially what happened was after the big bang, things interacted for a while and then everything just kind of [IB] into atoms and everything and then the light that was remaining from the big bang decoupled from everything else because now everybody is neutral so the light is just going to propagate through. The cosmic background radiation is the remnants today of that light that decoupled about 300,000 years after the beginning of the universe. Now when we measure the intensity of the different wavelengths that come in that cosmic background radiation, we find a perfect blackbody spectrum. In fact it's the most perfect one ever found in nature. Cosmic background radiation with a temperature of 2.725 kelvin. It's been cooling off since the big bang. And by now it's cooled to 2.725 kelvin.

Anyway, that's blackbody radiation spectrum, one of the most important things to come out of nineteenth century physics.